Abstract
The mathematical description of the vibration of a complex structure is very complicated because of the many modes of motion in which the structure can respond. However, we are primarily interested in the periodic vibrations that are excited by harmonically varying forces and in the building up and decay of such vibrations, since the response of a vibrator to unsteady forces can be deduced from its response to harmonic forces with the aid of Fourier analysis or the Laplace transform. The harmonic sine and cosine functions are very inconvenient because of the complex addition and multiplication theorems and the complexity of all the other theorems that apply to these functions. Fortunately, it is possible to eliminate sine and cosine completely by introducing rotating vectors as the primary variables. The sine and cosine can be defined as the projections of such a vector on the vertical and horizontal axis, respectively, both being plotted as functions of the angle of this vector with the horizontal axis according to the procedure shown in Fig. 1. A condensed notation can be introduced by replacing the sinusoidal functions by rotating vectors and by considering these rotating vectors as the primary variables. The harmonic functions can then be easily reconstructed from the rotating vectors.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bickley, W. C., Talbot, A.: An introduction to the theory of vibrating systems, Chapter III. Oxford Press 1961.
Brenner, E., Mansour, J.: Analysis of electric circuits. New York, N. Y.: McGraw-Hill 1967.
Churchill, R. V.: Complex variables and applications. New York, N. Y.: McGraw-Hill. 1960.
Close, Ch. M.: The analysis of linear circuits. New York, N. Y.: Hartcourt, Brace and World, Inc. 1966.
Crafton, P. A.: Shock and vibration in linear systems, Chapter I and II. New York, N. Y.: Harper Brothers.
Cruz, Y. B., Jr., Valkenburg, M. E. Van: Introductory signals and circuits. Waltham, MA: Blaisdell Publishing Company. 1967.
Den Hartog, J. P.: Mechanical vibrations, Chapter I. New York, N. Y.: McGraw-Hill. 1947.
Johnson, W. C.: Mathematical and physical principles of engineering analysis. New York, N. Y.: McGraw-Hill. 1944.
Kimbal, A. L.: Vibration prevention in engineering, Chapter V. New York, N. Y.: Wiley. 1946.
Skilling, H. H.: Electrical engineering circuits. New York, N. Y.: Wiley. 1965.
Skudrzyk, E. J.: Simple and complex vibratory systems. The Pennsylvania State University Press 1968.
Valkenburg, M. E. Van: Network analysis. Englewood Cliffs, N. J.: Prentice Hall. 1965.
Papers on Internal Friction(Two Constants of Internal Friction)
Liebermann, L. N.: The second viscosity of liquids. Phys. Rev. 75 (1949) 1415–1422 Erratasm 76 (1949) 770.
Skudrzyk, E.: Theorie der inneren Reibung in Gasen und Flüssigkeiten und die Schallabsorption. Acta Physica Austriaca 2 (1948) 148–181.
Skudrzyk, E.: Die innere Reibung und die Materialverluste fester Körper I. Osten.. Ing.-Arch. 3 (1949) 356–373.
Summaries of the Theory in:
Snowdon, J. C.: Vibration and shock in damped systems. New York, N. Y.: Wiley. 1968.
Skudrzyk, E.: Vibration of complex vibratory systems. University Park, Penna.: The Pennsylvania State University Press. 1968;
Skudrzyk, E.: Grundlagen der Akustik. Wien: Springer. 1954. ( Contains numerous references. )
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1971 Springer-Verlag/Wien
About this chapter
Cite this chapter
Skudrzyk, E. (1971). Complex Notation and Symbolic Methods. In: The Foundations of Acoustics. Springer, Vienna. https://doi.org/10.1007/978-3-7091-8255-0_3
Download citation
DOI: https://doi.org/10.1007/978-3-7091-8255-0_3
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-8257-4
Online ISBN: 978-3-7091-8255-0
eBook Packages: Springer Book Archive